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14.E: Chapter 6 Review Exercises

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Chapter Review Exercises

Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

12a2b3, 15ab2

Answer

3ab2

12m2n3,42m5n3

15y3, 21y2, 30y

Answer

3y

45x3y2, 15x4y, 10x5y3

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

35y+84

Answer

7(5y+12)

6y2+12y6

18x315x

Answer

3x(6x25)

15m4+6m2n

4x312x2+16x

Answer

4x(x23x+4)

3x+24

3x3+27x212x

Answer

3x(x29x+4)

3x(x1)+5(x1)

Factor by Grouping

In the following exercises, factor by grouping.

axay+bxby

Answer

(a+b)(xy)

x2yxy2+2x2y

x2+7x3x21

Answer

(x3)(x+7)

4x216x+3x12

m3+m2+m+1

Answer

(m2+1)(m+1)

5x5yy+x

Factor Trinomials

Factor Trinomials of the Form x2+bx+c

In the following exercises, factor each trinomial of the form x2+bx+c.

a2+14a+33

Answer

(a+3)(a+11)

k216k+60

m2+3m54

Answer

(m+9)(m6)

x23x10

In the following examples, factor each trinomial of the form x2+bxy+cy2.

x2+12xy+35y2

Answer

(x+5y)(x+7y)

r2+3rs28s2

a2+4ab21b2

Answer

(a+7b)(a3b)

p25pq36q2

m25mn+30n2

Answer

Prime

Factor Trinomials of the Form ax2+bx+cax2+bx+c Using Trial and Error

In the following exercises, factor completely using trial and error.

x3+5x224x

3y321y2+30y

Answer

3y(y5)(y2)

5x4+10x375x2

5y2+14y+9

Answer

(5y+9)(y+1)

8x2+25x+3

10y253y11

Answer

(5y+1)(2y11)

6p219pq+10q2

81a2+153a+18

Answer

9(9a1)(a+2)

Factor Trinomials of the Form ax2+bx+cax2+bx+c using the ‘ac’ Method

In the following exercises, factor.

2x2+9x+4

18a29a+1

Answer

(3a1)(6a1)

15p2+2p8

15x2+6x2

Answer

(3x1)(5x+2)

8a2+32a+24

3x2+3x36

Answer

3(x+4)(x3)

48y2+12y36

18a257a21

Answer

3(2a7)(3a+1)

3n412n396n2

Factor using substitution

In the following exercises, factor using substitution.

x413x230

Answer

(x215)(x2+2)

(x3)25(x3)36

Factor Special Products

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

25x2+30x+9

Answer

(5x+3)2

36a284ab+49b2

40x2+360x+810

Answer

10(2x+9)2

5k370k2+245k

75u430u3v+3u2v2

Answer

3u2(5uv)2

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

81r225

169m2n2

Answer

(13m+n)(13mn)

25p21

9121y2

Answer

(3+11y)(311y)

20x2125

169n3n

Answer

n(13n+1)(13n1)

6p2q254p2

24p2+54

Answer

6(4p2+9)

49x281y2

16z41

Answer

(2z1)(2z+1)(4z2+1)

48m4n2243n2

a2+6a+99b2

Answer

(a+33b)(a+3+3b)

x216x+64y2

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

a3125

Answer

(a5)(a2+5a+25)

b3216

2m3+54

Answer

2(m+3)(m23m+9)

81m3+3

General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

24x3+44x2

Answer

4x2(6x+11)

24a49a3

16n256mn+49m2

Answer

(4n7m)2

6a225a9

5u445u2

Answer

5u2(u+3)(u3)

n481

64j2+225

Answer

prime

5x2+5x60

b364

Answer

(b4)(b2+4b+16)

m3+125

2b22bc+5cb5c2

Answer

(2b+5c)(bc)

48x5y2243xy2

5q215q90

Answer

5(q+3)(q6)

4u5v+4u2v3

10m46250

Answer

10(m5)(m+5)(m2+25)

60x2y75xy+30y

16x224xy+9y264

Answer

(4x3y+8)(4x3y8)

Polynomial Equations

Use the Zero Product Property

In the following exercises, solve.

(a3)(a+7)=0

(5b+1)(6b+1)=0

Answer

b=15, b=16

6m(12m5)=0

(2x1)2=0

Answer

x=12

3m(2m5)(m+6)=0

Solve Quadratic Equations by Factoring

In the following exercises, solve.

x2+9x+20=0

Answer

x=4, x=5

y2y72=0

2p211p=40

Answer

p=52,p=8

q3+3q2+2q=0

144m225=0

Answer

m=512, m=512

4n2=36

(x+6)(x3)=8

Answer

x=2, x=5

(3x2)(x+4)=12

16p3=24p2+9p

Answer

p=0, p=34

2y3+2y2=12y

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, f(x)=x2+11x+20, ⓐ find when f(x)=8 ⓑ Use this information to find two points that lie on the graph of the function.

Answer

x=7 or \x=4
(7,8) (4,8)

For the function, f(x)=9x218x+5, ⓐ find when f(x)=3 ⓑ Use this information to find two points that lie on the graph of the function.

In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

f(x)=64x249

Answer

x=78 or x=78
(78,0), (78,0)(0,49)

f(x)=6x213x5

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive numbers is 399. Find the numbers.

Answer

The numbers are 21 and 19 or 19 and 21.

The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.

A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

Answer

The lengths are 8, 15, and 17 ft.

Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function h(t)=16t2+64t+80 models the height, h, of the ball above the ground as a function of time, t. Find: ⓐthe zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 80 feet above the ground. ⓒ the height the ball will be at t=2 seconds which is when the ball will be at its highest point.

Chapter Practice Test

In the following exercises, factor completely.

80a2+120a3

Answer

40a2(2+3a)

5m(m1)+3(m1)

x2+13x+36

Answer

(x+7)(x+6)

p2+pq12q2

xy8y+7x56

Answer

(x8)(y+7)

40r2+810

9s212s+4

Answer

(3s2)2

6x211x10

3x275y2

Answer

3(x+5y)(x5y)

6u2+3u18

x3+125

Answer

(x+5)(x25x+25)

32x5y2162xy2

6x419x2+15

Answer

(3x25)(2x23)

3x336x2+108x

In the following exercises, solve

5a2+26a=24

Answer

a=45, a=6

The product of two consecutive integers is 156. Find the integers.

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

Answer

The width is 12 inches and the length is 14 inches.

Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function h(t)=16t2+64t+80 models the height, h, of the ball above the ground as a function of time, t. Find: ⓐthe zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 128 feet above the ground. ⓒ the height the ball will be at t=4 seconds.

For the function, f(x)=x27x+5, ⓐ find when f(x)=7 ⓑ Use this information to find two points that lie on the graph of the function.

Answer

x=3 or x=4(3,7) (4,7)

For the function f(x)=25x281, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒthe y-intercept of the graph of the function.

Glossary

degree of the polynomial equation
The degree of the polynomial equation is the degree of the polynomial.
polynomial equation
A polynomial equation is an equation that contains a polynomial expression.
quadratic equation
Polynomial equations of degree two are called quadratic equations.
zero of the function
A value of xx where the function is 0, is called a zero of the function.
Zero Product Property
The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.

This page titled 14.E: Chapter 6 Review Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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